Are they the words of Elisabeth or the narrator?
Blessed is the one who believed that there will be a fulfillment to that which was spoken to her by the Lord!
There are several reasons to understand the speaker here as Elisabeth. But if you know the rest of the story, it is not difficult to hear an exhortation from the narrator to the reader. It doesn't have to be one or the other.
The way this verse is worded might make it difficult for some men to hear it as an exhortation. The blessing is for the one who believes what the Lord has spoke to her (αυτη). In this part of the story, however, Mary's response to the angel's good news stands in contrast to Zechariah's (Luke 1:18-20). Zechariah gets smitten; Mary gets the stage to perform the Magnificat.
Sure, these are Elisabeth's words to Mary, but Mary is being held up as an example. Blessed is the one who, like Mary, believes what the Lord has spoken. The Old Testament promised an eternal kingdom for a Davidic king. Luke's Gospel promises that Jesus of Nazareth is this king (Luke 1:26-33). After hearing the end of his story, Luke intends for the reader's heart to burn with confidence that this is true and is hope giving (Luke 24:30-35). Luke 1:45 hangs over the Gospel and directs the reader as to how she, or he, should respond.